A temporal sampling strategy for hydraulic tomography analysis

Authors


Abstract

[1] This paper investigates optimal sampling times of drawdowns for the analysis of hydraulic tomography (HT) survey. The investigation was carried out by analyzing the spatial and temporal evolution of cross-correlations between the head responses at an observation well and transmissivity (T) and storage coefficient (S) properties during a pumping test in homogeneous and heterogeneous aquifers. The analysis shows that the cross-correlation between the head and S values is limited to the region between the observation and the pumping well in the aquifers: It reaches the highest value near the early time (tm), and decays to zero afterwards. The time tm is approximately equal to the time t0 at which the extrapolated drawdown from the first straight line portion of an observed drawdown-log time plot becomes zero. At early times, the high cross-correlation between the head and T is confined to the region between the observation and the pumping well. This region then evolves into two humps: One on each side of the circular region encompassing the observation well and the pumping well. The size of the two humps expands and their values reach the maximum as flow reach steady-state. As a consequence, we hypothesize that pairs of head data at t0 and those at either the steady-state or a late time during an HT survey could yield the best estimates of the heterogeneous T and S fields. Results from numerical experiments have verified this hypothesis and demonstrated that this sampling strategy is generally applicable even when the boundary condition is unknown. We, therefore, recommend in principle that (1) carrying out pumping tests of HT surveys for sufficiently long period of time such that drawdown reaches the entire area of interest and (2) using a constant head or zero drawdown for all boundaries during the inverse modeling analysis.

1. Introduction

[2] Transmissivity (T) and storage coefficient (S) are two important properties that control groundwater flow in aquifers and are of practical importance for water resources development and management as well as protection and remediation of groundwater. After Yeh and Liu [2000] and Liu et al. [2002] sequential pumping tests, multiwell interference tests, or hydraulic tomography (HT) have been the subject of active theoretical, laboratory, and recently field research to characterize the spatial distributions of hydraulic parameters at a higher level of detail than traditional methods [e.g., Gottlieb and Dietrich, 1995; Yeh and Liu, 2000; Liu et al., 2002; Bohling et al., 2002, 2007; Brauchler et al., 2003, 2011; Zhu and Yeh, 2005, 2006; Liu et al., 2007; Ni and Yeh, 2008; Straface et al., 2007; Xiang et al., 2009; Illman et al., 2007-2011; Fienen et al., 2008; Kuhlman et al., 2008; Castagna and Bellin, 2009; Berg and Illman, 2011; Cardiff et al., 2009, 2012; Cardiff and Barrash, 2011; Huang et al., 2011; Li et al., 2005, 2008; Liu and Kitanidis, 2011; Yin and Illman, 2009]. Note that Cardiff and Barrash (2011) provided a summary of all peer-reviewed HT studies (1D/2D/3D).

[3] These studies showed that transient HT can identify not only the pattern of the heterogeneous hydraulic conductivity (K) or transmissivity (T) field, but also the variation of specific storage (Ss) or the storage coefficient (S) [see Zhu and Yeh, 2005, 2006; Liu et al., 2007; Xiang et al., 2009, in particular]. More importantly, they have demonstrated that the hydraulic property fields estimated by HT can yield much better predictions of flow and solute transport processes than other conventional characterization approaches [Illman et al., 2011]. Berg and Illman [2011], in particular, substantiated the robustness of HT for a highly heterogeneous geological medium with a variance of log hydraulic conductivity of 5.4 and a vertical correlation scale of 0.15 m.

[4] Nonetheless, many practical issues remain, including (1) the design of spatial sampling and pumping locations, (2) the duration and magnitude of the pumping rate during an HT survey, and (3) the frequency of temporal sampling. Specifically, is it necessary to conduct the pumping test to reach a steady-state? How many time/drawdown data in a well hydrograph should be used to obtain good estimates of T and S field while minimizing computational effort for the HT analysis? While Yeh and Liu [2000] conducted a preliminary investigation of the spatial sampling issues, the temporal sampling issues are opened for investigation.

[5] The sampling time requirements vary with the method of interpretation. For instance, Vasco et al. [2000], Vasco and Karasaki [2006], Brauchler et al. [2003, 2011], and He et al. [2006] developed methods based on travel time and amplitude of drawdown-time curves at different locations from the pumping tests conducted in tomographic format to estimate hydraulic properties. Similar to the travel time approaches, Li et al. [2005] and Zhu and Yeh [2006] developed temporal moment approaches, which are based on multidimensional flow model. These approaches either implicitly or explicitly require a complete well hydrograph to determine the travel time and the amplitude (i.e., the temporal moments).

[6] On the other hand, Yeh and Liu [2000], Fienen et al. [2008], and Liu and Kitanidis [2011] developed HT interpretation methods based on steady-state flow models which require only steady-state heads or drawdowns. Zhu and Yeh [2005] reported that heads are highly correlated in time during HT experiments, As a result, Zhu and Yeh [2005], Liu et al. [2007], Straface et al. [2007], Xiang et al. [2009], Illman et al. [2009], Castagna et al. [2011], Berg and Illman [2011], Cardiff and Barrash [2011], and Cardiff et al. [2012] analyzed transient HT data using only a small numbers (four or five) of selected heads or drawdowns of the drawdown-time well hydrograph. Bohling et al. [2002], however, used 51 drawdown-time data points of each well hydrograph of 42 sampling ports of 14 pumping tests (a total of 29,988 drawdown data points) for transient inversion for hydraulic conductivity field only, assuming the S field was known exactly. Because of the large amount of head data used, the amount of computational resource required was overwhelming. Consequently, they applied the concept of steady shape to their analysis of HT data for the K field. The term “steady shape” is used to designate conditions in an unsteady-state flow regime in which drawdown continues to change with time but the hydraulic gradient remains constant. Specifically, they suggested that the head differences between different observation points (hydraulic gradient) can be analyzed using steady-shape flow model during the steady-shape regime even though the drawdown remains transient. Hu et al. [2011] similarly used the steady-shape concept and travel time approach to analyze the HT survey. Bohling [2009] further claimed that the steady-shape approach reduces the influence of uncertainty in boundary conditions when compared with the approach that uses drawdowns in a steady-state flow model. Bohling et al. [2002], Bohling [2009], and Bohling and Butler [2010] also advocated that neither steady-shape nor transient approach for HT analysis was capable of revealing hydraulic conductivity variations outside of the region encompassed by the pumping and observation wells.

[7] This paper investigates the time of drawdown measurements of well hydrographs that can maximize the resolutions of T and S estimates in the analysis of a HT survey. We employed a first-order cross-correlation analysis to investigate the temporal and spatial evolutions of cross-correlation between the head at an observation well and T and S in homogeneous and heterogeneous aquifers during a pumping test. Based on the analysis, we explored the head information content at different time periods during a pumping test on spatial distribution of T and S, then proposed a temporal sampling strategy for HT analysis, and at last tested it using numerical examples with and without prior knowledge of the boundary conditions. At the end, the cross-correlation analysis is used to explain the robustness of HT on T estimation, even beyond well field, and also to explain the limitation of HT on S estimation.

2. Cross-Correlation Analysis

[8] The cross-correlation analysis presented in this paper assumes that the groundwater flow in two-dimensional, depth-averaged, saturated, heterogeneous, or homogeneous aquifers can be described by the following equations:

display math(1)

subject to boundary and initial conditions:

display math(2)

where H is the total head [L], x is the spatial coordinate (x={x, y}, [L]), Q(xp) is the pumping rate (L3/T/L3/) at the location xp, T(x) is the transmissivity [L2/T], and S(x) is the storage coefficient [–]. If the aquifer is conceptualized as homogenous, values of T(x) and S(x) are independent of x. H1 is the prescribed total head at Dirichlet boundary г1, q is the specific flux at Neumann boundary г2, n is a unit vector normal to the union of г1 and г2, and H0 represents the initial total head.

[9] These equations are necessary to derive the cross-correlation between the head or drawdown at an observation well and the transmissivity and storage coefficient fields of homogeneous and heterogeneous aquifers.

2.1. Homogeneous Aquifers

[10] We employ a probabilistic approach in order to analyze the uncertainty of the prediction of transient hydraulic head response to a pumping test in a homogeneous aquifer with uncertain T and S values, either due to measurement errors or lack of measurements. This approach assumes that the natural log of T (ln T) and that of S (ln S) are random variables, and math formula and math formula, where math formula and math formula are their means and math formula and math formula denote the perturbations. Note that logs of the parameters are dimensionless. Likewise, the uncertainty in the head is represented by math formula, where math formula is the mean and math formula is the perturbation. Expanding the transient hydraulic head in equation (1) in a Taylor series about the mean values of parameters, and neglecting second-order and higher order terms, the head perturbation at location x at a given time t can be expressed as:

display math(3)

where math formula [L] and math formula [L] are scalars and are the sensitivity of h at location x at a given time t with respect to change in ln T and ln S values of the homogeneous aquifer (scalars), respectively. Equation (3) states that the head perturbation at x and t is approximately a weighted sum of the perturbations, y and z. The weights are the sensitivity of h to y and that of h to z at (x, t).

[11] If the perturbations are casted into a probabilistic framework, and if the perturbation of ln T and that of ln S (i.e., y and z, respectively) are mutually independent from each other, the cross-covariance of h and y and that of h and z can be given as

display math(4)

where math formula and math formula are the variances of ln T and ln S, which represent the uncertainty associated with the ln T and ln S values of the homogeneous aquifer. The corresponding head variance at (x, t) based on equation (3) is given as:

display math(5)

[12] The cross-covariances, math formula [L] and math formula[L], are then normalized by the square root of the product of the variances of h(x, t) and ln T (i.e., math formula and math formula, respectively) or those of h (x, t) and ln S (i.e., math formula and math formula, respectively) to obtain their corresponding cross-correlations math formula and math formula at location x at time t.

display math(6)

[13] Note that these cross-correlations are dimensionless (ranging from −1 to +1) and represent the relationship between the uncertainty of a given parameter of the homogeneous aquifer and the uncertainty in the head at a given location and time due to uncertainty of all parameters. In effect, the cross-correlation between the head and a parameter is a product of a weighted sensitivity and uncertainty of the parameter.

[14] To analyze the cross-correlation between the hydraulic head at a given location and T and S parameters in unbounded homogeneous and isotropic aquifers, we use the Theis solution [Theis, 1935]. That is, the drawdown, math formula, at time t in an observation well at a radial distance r from a pumping well, which discharges at a constant rate Q, is given by

display math(7)

[15] Equation (7) leads to the following dimensionless sensitivities,

display math(8)
display math(9)

[16] Subsequently, the cross-correlation between h and ln T can be expressed as:

display math(10)

[17] The cross-correlation between h and ln S then becomes

display math(11)

[18] Note that in equations (10) and (11), we assume that math formula, where b is a constant.

[19] To investigate the effects of a straight-line impermeable or recharge boundary on these cross-correlations, we use the image well (superposition) approach [Bear, 1972]. That is, the drawdown or built up, math formula, at a distance r from the pumping or injection well can be formulated using the Theis solution as:

display math(12)

where math formula; math formula denotes the distance between the observation well and the image well; the minus sign in equation (12) is used for the recharge boundary condition and the plus sign is for the impermeable boundary. Accordingly, we obtain the dimensionless sensitivities of the head or drawdown with respect to change in ln T or ln S.

display math(13)
display math(14)

[20] Subsequently, we have the expressions for their corresponding cross-correlations:

display math(15)
display math(16)

[21] The cross-correlations between h and ln T and that between h and ln S (i.e., equations (15) and (16)) are plotted as a function of dimensionless time t* which equals to 1/u in three homogeneous aquifers in Figure 1a. They are (1) an unbounded aquifer with infinite lateral extents (equations (10) and (11)), (2) an aquifer with a line impermeable boundary (equations (15) and (16) with plus sign), and (3) an aquifer with a line recharge boundary (equations (15) and (16) with minus sign). In all these cases, the value of b is set to one. The observation well is assumed to be on the line perpendicular to the boundary and connecting the pumping well and the boundary. Effects of distances between the observation well and the boundary (i.e., 1r, 5r, and unbound) on the cross-correlations are also plotted in this figure. The distance 5r means that the distance between the boundary and the observation well is five times, the distance between the observation and pumping well, and so on.

Figure 1.

Cross-correlation between the head at an observation well and aquifer properties ln T (hy) and ln S (hz) as a function of dimensionless time for homogeneous aquifers with (a) different boundaries and (b) different b values (the ratio of variance in ln T to that of ln S). Unbnd denotes unbounded aquifers, Im stands for aquifers with an impermeable boundary, and Re stands for aquifers with a recharge boundary. r is the distance between the pumping well and observation. Im-5r or Re-5r denotes that the distance between the impermeable or recharge boundary and the observation is 5 times of r.

[22] As illustrated by the solid black and red lines (for the unbounded aquifer with infinite lateral extents) in Figure 1a, at early times the head at the observation well is positively correlated with ln S and negatively correlated with ln T. Physically, this means that if the observed head is high (or drawdown is small), the S value is likely to be large and the T value is small, and vice versa. At intermediate time (t* = 2.2) in the figure when the correlation of h and ln T is zero, the cross-correlation of the h and ln S reaches maximum. This dimensionless time can be converted to real time as:

display math(17)

[23] This time is approximately equal to the intercept t0 on time axis of a log time-drawdown plot of well hydrograph at any r, when the straight line portion of the plot is extrapolated to the point where the drawdown is zero according Cooper-Jacob's approach [Cooper and Jacob, 1946]. That is,

display math(18)

[24] Afterward, the value of math formula continuously grows and approaches some asymptotic value (close to 1) at large times, whereas the math formula value decreases rapidly and approaches some value (zero if steady-state condition is met). These results are consistent with the fact that at early time the head is controlled by both T and S and is only influenced by T near steady-state. Therefore, if the head in an observation well is high, the T of the aquifer must be high.

[25] As expected, the cross-correlations exhibit effects of either impermeable or recharge boundary once the drawdown reaches the boundary (see Figure 1a). In comparison with the cross-correlation between head and ln T of the unbounded aquifer, the presence of the recharge boundary accelerates the rise of the correlation and the impermeable boundary delays it. On the other hand, the recharge boundary accelerates the reduction of the correlation between head and ln S to zero, while the impermeable boundary postpones it. As shown in the figure, the dimensionless time where the maximum cross-correlation between head and ln S occurs is independent of the boundary location and type.

[26] Effects of the magnitude of b (ratio of math formula to math formula) on the cross-correlations are illustrated in Figure 1b for unbounded media. The larger the b value is, math formula starts at a more negative value, and the earlier it reaches one. On the other hand, the larger the b value is, the smaller the math formulavalue starts, the faster it rises, and the sharper it drops. The value of b does not alter the time to reach maximum math formula. These effects are also observed for aquifers with impermeable or recharge boundary.

2.2. Heterogeneous Aquifers

[27] To analyze the relationship between the head and ln T and that between the head and ln S values in a heterogeneous aquifer, we assume that ln T and ln S at every location of the aquifer are random variables with some spatial correlation (i.e., the ln T and ln S fields of the aquifer are collections of random variables: Stochastic processes or random fields). Again, math formula and math formula, where math formula and math formula are mean values and math formula and math formula denote perturbations, which represent spatial variability or uncertainty due to lack of measurements of these parameters. Likewise, the head is represented by math formula, where math formula is the mean and math formula is the perturbation caused by spatial variability or uncertainty of the parameters. The first-order approximation of the head perturbation at location math formula at a given time t is then given as:

display math(19)

where math formula and math formula are perturbation of ln T and ln S at location math formula and j=1,…, N, which is the total number of parameters in the aquifer (i.e., number of elements in a finite element domain); math formula and math formula are the sensitivity of h at location math formula at a given time t with respect to ln T and ln S perturbation at location math formula. Here, the Einstein's summation convention over the repeated suffix is used. In other words, the head perturbation at math formula is a weighted sum of perturbation of parameters ln T and ln S everywhere in the aquifer. The weights are the corresponding sensitivity values. Assuming ln T and ln S are mutually independent from each other, the cross-covariance matrices between h and y and between h and z are

display math(20)

[28]  math formula and math formula are covariance matrices of perturbation of ln T and ln S, which are modeled with the same exponential function using the same correlation scales Lx, Ly, and Lz in x, y, and z directions, respectively. The corresponding head covariance matrix based on equation (19) is given as:

display math(21)

[29] The superscript T denotes the transpose. The components of math formula at math formula are the head variances math formula, which represents the uncertainty in head at location math formula at a given time t due to the unknown heterogeneity in the aquifer. The cross-covariances, math formula and math formula, are then normalized by the square root of the product of the variances of h at math formula and ln T or those of h at math formula and ln S to obtain their corresponding cross-correlation math formula and math formula at location i and j at time t.

display math(22)

where math formula is the head variance at math formula and t; math formula and math formula are the variances of ln T and ln S, respectively. The cross-correlation (dimensionless) represents how the head perturbation at the location math formula at a given time t is influenced by the ln T or ln S perturbation at location math formula in a statistical sense. With a given mean T, S, and a pumping rate, these cross-covariances are evaluated numerically using the HT inverse model by Zhu and Yeh [2005] which is an extension of the earlier work by Yeh et al. [1996], Zhang and Yeh [1997], Li and Yeh [1999], Hughson and Yeh [2000], and Yeh and Liu [2000].

[30] The above cross-correlation analysis in heterogeneous aquifers is similar to the sensitivity analysis by Oliver [1993] but it adopts the stochastic or geostatistics concept. In particular, the cross-correlation analysis considers the variance (spatial variability) of the parameter and its spatial correlation structure (covariance function or variogram of the parameter) in addition to the most likely flow field, which is considered in the sensitivity analysis. Physically, the correlation structure represents the average dimensions of aquifer heterogeneity. The cross-correlation in essence represents the statistical relationship of spatial variability of a given parameter (T or S) at any location and the variability of head at an observation location in the aquifer. Note that the cross-correlation is the foundation of cokriging approach [e.g., Kitanidis and Vomvoris, 1983; Hoeksema and Kitanidis, 1984; Yeh and Zhang, 1996; Li and Yeh, 1999; Yeh et al., 1995], nonlinear geostatistical inverse approach [e.g., Kitanidis, 1995; Yeh et al., 1996; Zhang and Yeh, 1997; Hanna and Yeh, 1998; Hughson and Yeh, 2000], and HT inverse model [e.g., Yeh and Liu, 2000; Zhu and Yeh, 2005], geostatistical inverse modeling of electrical resistivity tomography [Yeh et al., 2002].

[31] To illustrate the behaviors of equation (22), a square-shaped synthetic 2-D confined aquifer (200 m × 200 m) was discretized into 100 × 100 square elements with 2 m in length and width. Four sides (i.e., east, west, north, and south) of the aquifer were assumed to have a prescribed hydraulic head of 100 m. An observation well was placed at x = 80 m, and y = 100 m and a pumping well was placed at x = 120 m, y = 100 m, where a constant discharge of 0.0006 m3/s was imposed. The geometric mean values of T and S of the aquifer are 0.000116 m2/s and S = 0.00014, respectively. The variance of ln T is 1.0 and variance of ln S is 0.2. Covariance functions of T and S are based on the exponential model with isotropic correlation scales in x and y directions that are 30 m. These properties of the aquifer and pumping rate are based on the field experiments reported by Way and McKee [1982].

[32] Contour maps of the cross-correlations between the head at an observation well and T perturbation everywhere in the domain (i.e., math formula) as well as the head field are plotted in Figures 2a and 2c for two early times (i.e., dimensionless time t* = 0.4 and 2.2), and Figures 3a and 3c for late time and steady-state (i.e., t* = 6.2 and 41.4), respectively. The dimensionless time t* which equals to 1/u was evaluated using the mean values of T and S. The corresponding cross-correlation maps for the head and the S field ( math formula) are shown in Figures 2b, 2d, 3b, and 3d. Note that the head fields are the means, which were obtained using the mean T and S values.

Figure 2.

Contour maps of cross-correlation between the head at the observation well (the white circle) and ln T (a and c) and ln S (b and d) everywhere in the aquifer at early dimensionless time 0.4, and 2.2 after pumping started at the pumping well (the black circle). The aquifer is surrounded by constant heads of 100 m. White contours are equipotential lines.

Figure 3.

Contour maps of cross-correlation between the head at the observation well (the white circle) and ln T (a and c) and ln S (b and d) everywhere in the aquifer at late dimensionless times 6.2, and 41.4 after pumping started at the pumping well (the black circle). The aquifer is surrounded by constant heads of 100 m. White contours are equipotential lines. Black lines with arrows are streamlines.

[33] At very early time (t* = 0.4, t = 200 s), the math formula values are significantly negative over a large area and are with the highest correlation (i.e., around −0.6) between the pumping well and observation well (Figure 2a). Meanwhile, the cross-correlation math formula ranges from 0.1 to 0.4 (Figure 2b). The high positive values (i.e., around 0.4) are confined to the small circular area between the pumping and the observation well. Within the aquifer domain, the nonzero velocities at this very early time are limited to the vicinity of the pumping well and the drawdown has not propagated to the observation well yet. As a result, the high cross-correlation at this time stage is not useful to relate hydraulic properties to the head at the observation well and beyond.

[34] At early time (t* = 2.2, t = 1100 s, approximately equal to t0 time), the noticeable drawdown have expanded to the observation well, but only to some limited areal extents (Figures 2c and 2d). The math formula values are close to zero everywhere except the small circular area centered between the pumping well and the observation well, where the values range from −0.5 to 0 (Figure 2c). The negative correlation means that if the head at the observation well in a heterogeneous aquifer is higher than the head calculated from the mean T and S values of the aquifer, the T values downstream of the observation are likely to be lower than the mean T value. Meanwhile, the math formula values are ranging from 0.1 to 0.4 over the area with the highest correlation are located in a small circular area between the pumping well and the observation well (Figure 2d). The positive cross-correlation implies that if the observed head in a heterogeneous aquifer is high in comparison with the head derived from the mean T and S values of the aquifer, the S values in the area between the pumping well and the observation well are likely to be higher than the mean S value. Note that the area of significant math formula values is much larger than that of math formula. These high cross-correlation values suggest that the head at the observation well at this time is highly influenced by the S heterogeneity within the small red circle with math formula = 0.4 enclosing the pumping and the observation wells although the area outside the circle also has some influence.

[35] At t* = 6.2 (intermediate time, t = 3000 s), as shown in Figures 3a and 3b, the edge of the cone of depression has reached the right-hand side of the constant head boundary. The shapes of math formula remain similar to that at t* = 2.2 but the values drop below 0.4 (Figure 3b), indicating that the effect of storage coefficient on head is diminishing and flow is moving toward a steady-state. At this time, math formula has evolved to positive values everywhere. The cross-correlation pattern forms two kidney-shaped humps: One at the left-hand (upstream) side of the observation well and the other at the right-hand (upstream) side of the pumping well (Figure 3a). The values in the area between the two wells are smaller.

[36] In order to explain the formation of this peculiar distribution, we will examine the cross-correlations along two streamlines opposite to each other since the flow is radial symmetrical and close to steady-state. The first streamline is the one starting from the left constant head boundary and passing through the observation well to the pumping well. The second one is the streamline starting from the right-hand side constant head boundary to the pumping well directly opposite to the first streamline. We will discuss the variation of math formula along the first streamline first. If the flow to the pumping well approaches a steady condition, and the head at the observation well is higher than the head at the same location that is simulated with the mean T, the average hydraulic gradient along the streamline upstream from the observation well is smaller than the mean gradient of the homogeneous medium. The T values upstream, therefore, are most likely to be higher than the mean T, and the T values along the same streamline downstream from the observation well to the pumping well are likely to be smaller. If the T values between the observation and pumping well are small on average, then the contribution to the pumping well from this streamline is likely to be smaller than that calculated using the mean T. As a result, the contribution from the second streamline to the pumping well must increase, and the T values, on average, along the second streamline upstream from the pumping well are in turn greater than that based on the mean T in order to sustain the well discharge.

[37] For this radial flow field, all streamlines converge to the pumping well. The cross-correlation distribution between head at observation well and ln T along each streamline follows the same principle as discussed above. However, such a cross-correlation pattern along other streamlines diminishes as the distance between these streamlines and the streamline that passes through the observation well increases. As a result, two kidney-shaped humps of high correlation values (one near the observation well and one near the pumping well) form.

[38] When the flow state reaches the steady-state (t* = 41.4 or t = 20,000 s), the shape of the spatial distribution of math formula values are similar to that at t* = 6.2 (Figure 3a) but their magnitudes everywhere in the aquifer are elevated (Figure 3c). Meanwhile, the math formula values diminish to zero everywhere reflecting the fact that the steady hydraulic head is not related to S (Figure 3d). Note that this two kidney-shaped spatial pattern of the cross-correlation between ln T and the head is different from the sensitivity behavior based on radial symmetric flow model as used by Bohling et al. [2002], which indicates that the head is only sensitive to T values in the region between the observation well and the pumping well.

[39] As shown in equation (22), the cross-correlations will be affected by the covariance functions of the parameters. When the correlation scales in both directions decrease to 2 m (one element size) (i.e., the parameter fields become uncorrelated random fields), the spatial patterns of the cross-correlation functions are identical to that of the sensitivity results by Oliver [1993] and Leven and Dietrich [2006], with the exception that they are normalized. On the other hand, when the correlation scales become much larger than the domain size, the temporal behavior of the cross-correlation functions becomes that for homogeneous case as shown in Figures 1a and 1b. These results in Figure 3 are similar to those in Figure 10 of Wu et al. [2005].

[40] These spatial patterns of math formula and math formula provide some insight to the limitation of HT for S estimation and its robustness for T estimation. Specifically, heterogeneities at the same correlation contour have the same amount of contribution to the head responses at the observation well. The radial symmetrical pattern of math formula (Figures 2b, 2d, 3b, and 3d) thus indicates that contributions of an S anomaly in the aquifer will be the same for any pairs of a pumping and an observation well that are separated by the same distance and centered at the same location. Hence, it is difficult to identify the S anomaly unless data from pairs of wells with different separation distances are available. Furthermore, the high resolution of the S estimates is only observed within the high correlation circular area enclosed by the two wells.

[41] The spatial patterns of math formula evolve from a symmetric pattern at early times, similar to those of math formula to radial nonsymmetric two-hump patterns at late times. The highest math formula values are at the two humps at the upstream of the observation and the pumping well. This unique two-hump pattern implies that if the location of one of the two wells (i.e., the flow) is altered and the heterogeneity at the hump associated with the new location is different from that at the previous location, the head at the observation well (regardless if its position is changed or not) will be different and will carry new information about heterogeneity. Furthermore, math formula values over entire area of cone of depression are relatively high at late time in comparison with other times. Therefore, adding more head data at late time or steady-state from new observation or pumping locations in the inverse modeling process helps to decipher the spatial distribution of T anomalies, and thus yield a better resolution of the spatial distribution of heterogeneity. This is the reason that joint interpretation of sequential pumping tests or multiwell pumping tests are superior to inverse modeling using one pumping test as demonstrated by Huang et al. [2011]. The flow-dependent nature of math formula (or the flow independence nature of math formula) also explains that HT is a more robust approach for estimating T than S, and HT can depict heterogeneity patterns not only within the well filed but also far away from the well field although at low resolutions.

3. Temporal Sampling Strategy for HT analysis

[42] According to temporal and spatial distributions of the cross-correlations, and head fields presented in the previous section, we conclude that the head observed at the given observation well at t* = 2.2 (or t0) carries the highest level of information about S perturbations over a large area of influence. The head data at this time is highly desirable for estimating S distribution. On the other hand, at late time or steady-state, the observed head at the given observation location carries the greatest level of information about T perturbations over a larger portion of the aquifer in comparison with the head at other time periods. Furthermore, the b values in equations (15) and (16) are generally greater than 10 or above based on our knowledge of T and S values for most aquifers. As indicated in Figure 1b, for these b values, the correlation between the head and S at these values drops rapidly after t0. There is only a very small time window where the drawdown is more correlated with S than T, and for most of the time during a pumping test, drawdown in most aquifers is dominated by T values. Hence, it is most appropriate to choose the head data at steady-state or late times to estimate the distribution of T, using HT. Combining these two observations, we hypothesize a temporal sampling strategy for interpretation of HT. That is, pumping tests of HT should last sufficiently long such that drawdown reaches the entire area of interest. HT analysis should use head data at the early time (t0) and late time (or steady-state) for the estimation of T and S to maximize the power of HT as well as to reduce computational expenses.

[43] To test this hypothesis which is built upon the stochastic ensemble concept and the first-order analysis, we created a synthetic, two-dimensional confined aquifer. This confined aquifer has the same dimension, numerical discretization, and spatial statistics describing heterogeneity of T and S of the aquifers used in section 2.2. However, the right-hand side of this aquifer is an impermeable boundary and all others three boundaries are constant head of 100 m. One realization of 10,000 pairs of T and S values of the aquifer were generated by the spectral method [Gutjahr, 1989] using the spatial statistics in section 2.2. The T and S fields were assumed to be independent. The generated T and S fields are shown in Figures 4a and 4b, respectively.

Figure 4.

The true T and S fields of the synthetic aquifer are shown in Figures 4a and 4b, respectively. White circles are the nine wells used in HT and dashed white lines delineate the four zones for HT performance evaluation. The estimated T and S field from the HT analysis with the correct impermeable boundary on the east side using the pair of head data at t0 and steady-state are illustrated in Figures 4c and 4d. The estimated T and S fields using a wrong boundary condition (constant head) on the east side are shown in Figures 4e and 4f. The impermeable boundary is manifested as low permeable zones in the estimated T field in Figure 4e.

[44] Nine wells were placed in the aquifer. The coordinates for the nine wells from 1 to 9 (small circles in Figures 4a and 4b) are (80 m, 120 m), (100 m, 120 m), (120 m, 120 m), (80 m, 100 m), (100 m, 100 m), (120 m, 100 m), (80 m, 80 m), (100 m, 80 m), and (120 m, 80 m), respectively. A pumping test was simulated at one of the five wells (wells 1, 3, 5, 7, and 9) with a constant pumping rate 0.0006 m3/s (51.84 m3/d) for 70,000 s with a time step of 5 s, and the head responses at the other eight wells were monitored. This simulation of the pumping test was repeated for the other wells until all five wells were considered. As a result, there are five pumping tests and each test has eight observed drawdown-time curves (Figure 5). For this heterogeneous aquifer, the flow field has become steady at nearly 30,000 s. Using the eight drawdown-time curves for each pump test, eight t0 values of Cooper-Jacob straight-line method were obtained for each one of the pumping tests. Since the distance between observation and pumping wells are different and the aquifer is heterogeneous, t0 time for each drawdown-log time curve is different and they range around 160–700 s. The reason we use t0 instead of tm (equation (17)) are the following: (1) the difference between the two is small, (2) tm requires the knowledge of the mean T and S, which are unknowns themselves, and (3) t0 can be obtained conveniently from well hydrographs of each individual well.

Figure 5.

Drawdown time curves of five pumping tests. In the legend, the first digit denotes the pumping well number and the second digit represents that of the observation well.

[45] Drawdowns (i.e., heads) at some selected times of simulated drawdown-time curves of the HT survey (Figure 5) were sampled to test our proposed strategy for estimating 10,000 pairs of T and S values of the aquifer. These selected time intervals include t0, 1000 s, 2000 s, 3000 s, 5000 s, 8000 s, and steady-state. These head data are noise-free since noise effects on HT interpretations have been investigated extensively by Xiang et al. [2009] and Mao et al. [2013]. The principle of reciprocity [Bruggeman, 1972] was not considered in the HT analysis even though it can reduce the analysis time.

[46] The interpretation or inverse modeling using selected head data were carried out by employing Simultaneous Successive Linear Estimator (SimSLE) [Xiang et al., 2009]. Instead of incorporating data sequentially into the estimation as in SSLE [Zhu and Yeh, 2006], SimSLE includes all selected drawdown data from different pumping tests during an HT survey simultaneously to estimate hydraulic properties of aquifers. A detailed description of the SimSLE can be found in Xiang et al. [2009].

[47] To evaluate the estimates, scatter plots of the true vs. estimated T and S values for each case are plotted and a linear model is then fitted to each case without forcing the intercept to zero. The slope and intercept of the fitted linear model, the coefficient of determination (R2), the mean absolute error (L1), and the mean square error (L2) norms are then used for evaluation since a single criterion is not sufficient. The L1 and L2 norms of T and S are computed as:

display math(23)

where N is the total element of the model, i indicates the element number, math formula is the estimated T or S value for the ith element, and math formula is the true T or S value of the ith element.

[48] In order to further evaluate the influence of boundary on the estimates, and to examine the resolution of HT estimates at different distances away from well field, we divided the aquifer into four zones using white dashed lines. As illustrated in Figure 4a, zone 1 represents the smallest square area, which is the area surrounded by the nine wells (well field); zone 2 is the next large square; while zone 3 denotes the area surrounded by the line, which includes zones 1 and 2; and zone 4 is the entire aquifer in Figure 4a.

[49] Three cases using head data at the five different sampling times as discussed previously were examined. Unless otherwise specified, analysis of all cases used the known geometric means, variances, and correlation scales of the T and S fields as the inputs to SimSLE. The effects of uncertainty of these parameters on HT interpretation are small as reported by Yeh and Liu [2000]. Their finding has also been verified by HT applications to many sandbox and field experiments over the past decade.

3.1. Case 1

[50] Five pairs of T and S fields were estimated using head data at the five pairs of sampling times: (1) 1000 s + t0, (2) 3000 s + t0, (3) 5000 s + t0, (4) 8000 s + t0, and (5) steady-state + t0. Again, t0 time is different for each observed well hydrographs during HT.

[51] Evaluation metrics (L1, L2, R2, and slope) of the estimated T fields in the four zones (different colors) are plotted for the five sampling time pairs (i.e., 1, 2, 3, 4, and 5 on the horizontal axis) in Figures 6a–6d, respectively. As indicated by the four metrics, the pair of t0 and late time data always results in a better of T field for all four zones than the pair of t0 and the early time. These figures also show that the estimated T fields in zone 1 (area within the well field) using the five different sampling time pairs are consistently better than zones 2, 3, and 4. Note that the improvement of T estimates in zone 1 tails off but improvements in outer zones continue although slightly as head data at later times were used.

Figure 6.

(a) L1, (b) L2, (c) R2, and (d) slope norms for estimated T field within the four zones, respectively, using head data at five pairs of sampling times: 1 (t0 + 1000 s), 2 (t0 + 3000 s), 3 (t0 + 5000 s), 4 (t0 + 8000 s), and 5 (t0 + steady). Head data at t0 time paired with head data at later time yield better T estimates for the four zones.

[52] Plots of the evaluation metrics (L1, L2, R2, and slope) of the estimated S fields in the four zones (different colors) for the five sampling time pairs (i.e., 1, 2, 3, 4, and 5 on the horizontal axis) are given in Figures 7a–7d, respectively. According to these figures, the pair of head data at late time and those at t0 also leads to better S estimates for the four zones than the pair of the early time and t0. Overall, these metrics consistently indicate that the S estimates in zone 1 (area within the well field) are better than zone 2, and so on, although there are some deviations (e.g., Figure 7c). Furthermore, results shown in both Figures 6 and 7 indicate that the pairs of head data at steady-state with those at t0 yields best estimates of T and S fields for all zones.

Figure 7.

(a) L1, (b) L2, (c) R2, and (d) slope norms for estimated S field within the four zones, respectively, using head data at five pairs of sampling times: 1 (t0 + 1000 s), 2 (t0 + 3000 s), 3 (t0 + 5000 s), 4 (t0 + 8000 s), and 5 (t0 + steady). Head data at t0 time paired with head data at later time yield better S estimates for the four zones.

3.2. Case 2

[53] To further confirm the hypothesis that the maximum cross-correlation between head and S at t0 can lead to better estimate of S field, head data from three pairs of sampling times, that is, (1) steady-state and t0, (2) steady-state and 1000 s, and (3) steady-state and 2000 s, were employed to estimate three sets of S fields. In this case, we first used steady-state head data and the steady HT code to estimate T field. Then, we treated the estimated T fields as known, and proceeded to estimation of three S fields using head data at t0, 1000 s, and 2000 s, respectively, using the transient HT code. Plots similar to those in Figures 6 and 7 are shown in Figure 8. All evaluation metrics (Figures 8a–8d) confirm that data from t0 and T estimates from the steady-state yield the best estimated S fields for all four zones (zone 1, in particular). The metrics for different zones fluctuate a little for times 2 and 3 but they all are less satisfactory in comparison with time 1. These results further substantiate the fact that the high cross-correlation between head and S at t0 time is indeed useful for estimating the heterogeneous S field.

Figure 8.

(a) L1, (b) L2, (c) R2, and (d) slope norms for estimated S field within the four zones, respectively, using head data at three pairs of sampling times:1 (steady + t0), 2 (steady + 1000 s), and 3 (steady + 2000 s). Head data at t0 time paired with head data at later time yield better T estimates for the four zones. Using T estimates from steady head data as known, head data at t0 yield the best S estimates in comparison with other times.

[54] The estimated T and S fields using the suggested sampling strategy (i.e., head data from t0 and steady-state) for HT analysis are illustrated in Figures 4c and 4d, respectively. In comparison with the true fields in Figures 4a and 4b, these estimated fields are much smoother (i.e., at lower resolutions) than the true fields. This is attributed to the fact that only nine wells were used for HT and the nature of the conditional expectation from SimSLE [Yeh and Liu, 2000]. Nevertheless, the general patterns of dominant heterogeneity in T and S (not limited to the area encompassed with the wells) have merged in the estimates. In particular, the low permeable regions along the upper-right boundary and at the left center of the aquifer (left of well #4) as well as the high permeable zone near well #9 stand out in the estimated T field. Likewise, the low and high S regions on the left-side and right-side of the aquifer are also depicted. Generally, the S estimates are not as vivid as in the T estimates, as explained before using the cross-correlation analysis.

[55] Contrary to the claims by Bohling et al, [2002], Bohling [2009], and Bohling and Bulter [2010], the above results unequivocally demonstrate that HT can reveal T and S heterogeneity outside the well field, as elucidated by our cross-correlation analysis, and as supported by the results of Huang et al. [2011], Berg and Illman [2011], Liu and Kitanidis [2011], Xiang et al. [2009], Liu et al. [2007], and Liu et al [2002]. Note these estimates are the conditional effective T and S fields, which are the most likely and unbiased T and S fields that can reproduce observed heads of the HT survey, with the given limited information.

[56] These results support our proposed sampling hypothesis: (1) pumping tests of HT should last sufficiently long such that drawdown reaches the entire area of interest and (2) HT analysis should use head data at the early time (t0) and late time (or steady-state) for the estimation of T and S to maximize the power of HT as well as to reduce computational expenses. However, these results are based on the assumption that the exact boundary conditions are known.

3.3. Case 3

[57] In order to investigate impacts of a wrong boundary on the estimates based on our suggested sampling strategy, we interpreted the same pair data at steady-state and t0 time of Case 1 using a constant head boundary condition for the four sides of the aquifer (i.e., the right-hand side impermeable boundary was replaced by a constant head of 100 m, the same as the other three.) The estimated T and S fields based on this wrong boundary condition are illustrated in Figures 4e and 4f, respectively. Comparing estimates based on incorrect east side boundary with the true fields (Figures 4a and 4b), we see that even though the wrong boundary condition was used, HT using SimSLE still yield the general patterns of T and S estimates similar to the true ones. The general patterns of T and S estimates based on the incorrect boundary are also similar to those estimates based on the data set with correct boundary conditions (Figures 4c and 4d). However, the high T zones (greenish zones in Figure 4c) in the vicinity of the east-side impermeable boundary are replaced by low permeable zones (blue zones in Figure 4e) to reflect the presence of the impermeable boundary. The pattern of the estimated S field based on the incorrect boundary condition remains similar to that based on the correct boundary condition, except the values of S on the right-hand side increases greatly due to the introduction of the low permeable zones in the region.

[58] Based on these results, we may conclude that HT analysis using the SimSLE, and the drawdown data at t0 time and steady-state yields reasonable estimates of the general patterns of T and S fields even without any prior knowledge of the boundary conditions. Specifically, using the assumption of constant head boundaries to interpret HT surveys in an aquifer with an impermeable boundary, the true T field near the impermeable boundary was identified as low T zones. On the negative side, the true T field adjacent to the boundary was incorrectly estimated to reflect the impermeable boundary. On the other hand, we may argue that the likely location of the impermeable boundary was identified. As a matter of fact, this result also supports the interpretation by Straface et al. [2007] of an HT analysis of a field sequential pumping test experiment. They attributed the low permeability zones identified by HT, which surround the modeling domain, to possible pinch-out of the confined aquifer based on the local geological setting.

[59] Overall, results of our cross-correlation analysis and numerical experiments challenge those reported by Bohling et al. [2002], Bohling [2009], and Bohling and Butler [2010]. We agree with Bohling and Butler [2010] that an infinite number of possible T and S fields could match all the head data from the sequential pumping tests. We emphasize that nonunique or uncertain solutions could exist in both forward and inverse modeling problems unless the problem is well-defined with necessary conditions [Yeh et al., 2011; Mao et al., 2013]. HT is an approach to collect data from an existing well field to reduce the uncertainty. We attribute the cautionary notes about limitations of HT by Bohling and Butler [2010] and Butler [2008] to their choice of forward model (axial symmetric radial flow model), and inverse methodology (i.e., pilot point approach) as explained by Huang et al. [2011].

[60] At last, while the cross-correlation analysis is built upon the ensemble concept (describing the general behaviors) and the first-order analysis (small perturbations), applications of HT to an aquifer with a single realization of T and S fields with relatively large variability confirm its usefulness for providing insights to the HT analysis.

4. Summary and Conclusions

[61] Selection of drawdown data at appropriate time intervals is important to joint interpretation of HT tests. Based on the analysis of the spatial and temporal distributions of the cross-correlation between head observed at a well and ln T and ln S everywhere in the aquifer, and drawdown distributions, we recommend that pumping tests of HT should last sufficiently long such that drawdown reaches the entire area of interest. Head data at the early time (t0) and late time or steady-state should be used for the estimation of T and S during the HT analysis. The t0 time is the time at which the drawdown extrapolated from first straight line portion of an observed drawdown-log time plot becomes zero, according to the Jacob straight-line method. At this time, the cross-correlation between h and S is the highest over a large portion of the aquifer. On the other hand, the head at an observation well is highly correlated with ln T heterogeneity over a large portion of the aquifer at the late time, steady or near steady-state. Note that longer pumping tests in HT surveys allow detection of heterogeneity at great distances at low resolutions but do not improve the estimate within the well field significantly. Ultimately, resolutions of the HT estimates depend highly on the number of monitoring wells [Yeh and Liu, 2000; Liu et al., 2002].

[62] If the flow regime approaches steady-state conditions, one should estimate T field first using the steady-flow inverse approach and then S field by the transient flow analysis using drawdowns at t0. We also recommend the use of constant head (or zero drawdown) boundaries for the interpretation of the HT data. Effects of impermeable boundaries manifested in head measurements will be reflected in the estimated T field using the SimSLE.

[63] The cross-correlation analysis also offers insights into the robustness of HT for estimating the T distribution and its limitation for estimating S. Specifically, the radially symmetric pattern of the cross-correlation of the head at a well and S heterogeneity everywhere in the aquifer during pumping at the pumping well attributes to the difficulty in delineating the heterogeneity of S even data from more pairs of wells are available. On the other hand, the two kidney-shaped humps near the observation well and the pumping well of the cross-correlation of head and T heterogeneity suggest that if the location of one of a pair of pumping and observation wells is changed and the heterogeneity near the well is not the same, the observed head contains new information. This explains the scenario-dependent nature of T and S estimates based on single pumping test experiment discussed by Huang et al. [2011]. Furthermore, it explicates the rationale behind sequential pumping test (HT, or multiwell interference pumping tests), and the reason that joint interpretation of multiwell pumping tests are superior to inverse modeling efforts using one pumping test and the same number of observation wells. At last, our results appears to support the call for change the way we collect and analyze data for characterizing aquifers by Yeh and Lee [2007] and promote the concept for cat-scanning the groundwater basin using natural stimuli as proposed by Yeh et al. [2008].

Acknowledgments

[64] This research was carried out during the first author's visit at the Department of Hydrology and Water Resources at the University of Arizona. The first author acknowledges supported by the National Natural Science Foundation of China (41102155, 41272258), National Basic Research Program of China (2010CB428802), the Fundamental Research Fund for National University, China University of Geosciences (Wuhan) (CUGL090215, CUG120113), open research fund of State Key Laboratory of Geohazard Prevention and Geoenvironment Protection (Chengdu University of Technology) (GZ2006-07). The second and third authors acknowledge support of NSF grant EAR-1014594, the support of “China Scholarship Council.” Supports from Jilin University, Jilin, China, and a grant from ESTCP subcontract through AMEC are also acknowledged. Yonghong Hao acknowledges funding from the National Natural Science Foundation of China 40972165. The authors would also like to give special thanks to Jeff Gawad and Michael Tso for proofreading the manuscript. Finally, the authors are also grateful to the editor, the AE, and the four reviewers for their encouraging, insightful and constructive comments.